To analyze each proposition step by step:

Proposition A: Any three vectors in space can be used as a basis.
- A basis for a space consists of linearly independent vectors that span the space.
- Three vectors can span a three-dimensional space if and only if they are non-coplanar.
- Therefore, the statement is not entirely correct because it omits the condition that the vectors must be non-coplanar.
- Conclusion: Proposition A is incorrect.

**Proposition B: Given vectors $\overrightarrow{a}\bot \overrightarrow{b}$, then $\overrightarrow{a}$ and $\overrightarrow{b}$ cannot form a basis for space with any other vector.**
- Orthogonality does not prevent two vectors from contributing to a basis. In fact, orthogonal vectors are inherently linearly independent.
- If a third vector $\overrightarrow{c}$ is neither parallel to nor lies in the plane formed by $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ can form a basis.
- Therefore, the statement in Proposition B is incorrect.

**Proposition C: Given vectors $\overrightarrow{a} \parallel \overrightarrow{b}$, then $\overrightarrow{a}$ and $\overrightarrow{b}$ cannot form a basis for space with any other vector.**
- Parallel vectors are linearly dependent.
- The addition of any other vector $\overrightarrow{c}$ cannot compensate for this linear dependency to form a complete basis for three-dimensional space.
- Hence, Proposition C correctly states that $\overrightarrow{a}$ and $\overrightarrow{b}$ cannot form a basis with any other vector.

**Proposition D: $A$, $B$, $M$, $N$ are four points in space. If $\overrightarrow{BA}$, $\overrightarrow{BM}$, $\overrightarrow{BN}$ cannot form a basis for space, then $A$, $B$, $M$, $N$ are coplanar.**
- The inability of $\overrightarrow{BA}$, $\overrightarrow{BM}$, $\overrightarrow{BN}$ to form a basis implies these vectors are not linearly independent.
- This lack of linear independence indicates that the points $A$, $B$, $M$, $N$ lie in a single plane, thus being coplanar.
- Proposition D is correct in stating the relationship between the vectors not forming a basis and the coplanarity of the points.

Final Conclusion:
The correct propositions are $C$ and $D$.

Hence, the answer encapsulated is $\boxed{C \text{ and } D}$.